An Itô formula for a fractional Brownian sheet with arbitrary Hurst parameters

By using the white noise theory for a fractional Brownian sheet, we derive an Itô formula for the fractional Brownian sheet with arbitrary Hurst parameters .

     

Images of the Brownian sheet

An -parameter Brownian sheet in maps a non-random compact set in to the random compact set in . We prove two results on the image-set :

(1) It has positive -dimensional Lebesgue measure if and only if has positive -dimensional capacity. This generalizes greatly the earlier works of J. Hawkes  (1977), J.-P. Kahane  (1985), and Khoshnevisan (1999).

(2) If , then with probability one, we can find a finite number of points such that for any rotation matrix that leaves in , one of the 's is interior to . In particular, has interior-points a.s. This verifies a conjecture of T. S. Mountford  (1989).

This paper contains two novel ideas: To prove (1), we introduce and analyze a family of bridged sheets. Item (2) is proved by developing a notion of ``sectorial local-non-determinism (LND).' Both ideas may be of independent interest.

We showcase sectorial LND further by exhibiting some arithmetic properties of standard Brownian motion; this completes the work initiated by Mountford (1988).

     

Entropy-constrained functional quantization of Gaussian processes

The sharp asymptotics for the entropy-constrained -quantization errors of Gaussian measures on a Hilbert space and in particular, for Gaussian processes is derived. The condition imposed is regular variation of the eigenvalues of the covariance operator.

     

Brownian sheet images and Bessel-Riesz capacity

We show that the image of a 2-dimensional set under -dimensional, 2-parameter Brownian sheet can have positive Lebesgue measure if and only if the set in question has positive ()-dimensional Bessel-Riesz capacity. Our methods solve a problem of J.-P. Kahane.

     

Asymptotics for the heat equation in the exterior of a shrinking compact set in the plane via Brownian hitting times

Let and let be a continuous, nonincreasing function on satisfying . Consider the heat equation in the exterior of a time-dependent shrinking disk in the plane:

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If there exist constants and a constant 0$"> such that , for sufficiently large , then . The same result is also shown to hold when is replaced by , where . Also, a discrepancy is noted between the asymptotics for the above forward heat equation and the corresponding backward one. The method used is probabilistic.

     

A path-transformation for random walks and the Robinson-Schensted correspondence

The author and Marc Yor recently introduced a path-transformation with the property that, for belonging to a certain class of random walks on , the transformed walk has the same law as the original walk conditioned never to exit the Weyl chamber . In this paper, we show that is closely related to the Robinson-Schensted algorithm, and use this connection to give a new proof of the above representation theorem. The new proof is valid for a larger class of random walks and yields additional information about the joint law of and . The corresponding results for the Brownian model are recovered by Donsker's theorem. These are connected with Hermitian Brownian motion and the Gaussian Unitary Ensemble of random matrix theory. The connection we make between the path-transformation and the Robinson-Schensted algorithm also provides a new formula and interpretation for the latter. This can be used to study properties of the Robinson-Schensted algorithm and, moreover, extends easily to a continuous setting.

     

Small Ball Asymptotics for the Stochastic Wave Equation

We examine the small ball asymptotics for the weak solution X of the stochastic wave equation

Efficient computation of root numbers and class numbers of parametrized families of real abelian number fields

Let be a parametrized family of simplest real cyclic cubic, quartic, quintic or sextic number fields of known regulators, e.g., the so-called simplest cubic and quartic fields associated with the polynomials and . We give explicit formulas for powers of the Gaussian sums attached to the characters associated with these simplest number fields. We deduce a method for computing the exact values of these Gaussian sums. These values are then used to efficiently compute class numbers of simplest fields. Finally, such class number computations yield many examples of real cyclotomic fields of prime conductors and class numbers greater than or equal to . However, in accordance with Vandiver's conjecture, we found no example of for which divides .

     

The exponent three class group problem for some real cyclic cubic number fields

We determine all the simplest cubic fields whose ideal class groups have exponent dividing , thus generalizing the determination by G. Lettl of all the simplest cubic fields with class number and the determination by D. Byeon of all all the simplest cubic fields with class number . We prove that there are simplest cubic fields with ideal class groups of exponent (and simplest cubic fields with ideal class groups of exponent , i.e. with class number one).

     

Higher order PDE's and iterated processes

We introduce a class of stochastic processes based on symmetric -stable processes, for . These are obtained by taking Markov processes and replacing the time parameter with the modulus of a symmetric -stable process. We call them -time processes. They generalize Brownian time processes studied in Allouba and Zheng (2001), Allouba (2002), (2003), and they introduce new interesting examples. We establish the connection of -time processes to some higher order PDE's for rational. We also obtain the PDE connection of subordinate killed Brownian motion in bounded domains of regular boundary.

     

Small ball probabilities for smooth Gaussian fields and tensor products of compact operators

We find the logarithmic L₂‐small ball asymptotics for a class of zero mean Gaussian fields with covariances having the structure of “tensor product”. The main condition imposed on marginal covariances is slow growth at the origin of counting functions of their eigenvalues. That is valid for Gaussian functions with smooth covariances. Another type of marginal functions considered as well are classical Wiener process, Brownian bridge, Ornstein–Uhlenbeck process, etc., in the case of special self‐similar measure of integration. Our results are based on a new theorem on spectral asymptotics for the tensor products of compact self‐adjoint operators in Hilbert space which is of independent interest. Thus, we continue to develop the approach proposed in the paper 6 , where the regular behavior at infinity of marginal eigenvalues was assumed.     

Tame kernels of cubic cyclic fields

There are many results describing the structure of the tame kernels of algebraic number fields and relating them to the class numbers of appropriate fields. In the present paper we give some explicit results on tame kernels of cubic cyclic fields. Table 1 collects the results of computations of the structure of the tame kernel for all cubic fields with only one ramified prime

In particular, we investigate the structure of the 7-primary and 13-primary parts of the tame kernels. The theoretical tools we develop, based on reflection theorems and singular primary units, enable the determination of the structure even of 7-primary and 13-primary parts of the tame kernels for all fields as above. The results are given in Tables 2 and 3.

     

Extremes of Gaussian random fields with regularly varying dependence structure

Let \(X(t), t\in \mathcal {T}\) be a centered Gaussian random field with variance function σ ²(?) that attains its maximum at the unique point \(t_{0}\in \mathcal {T}\), and let \(M(\mathcal {T})=\sup _{t\in \mathcal {T}} X(t)\). For \(\mathcal {T}\) a compact subset of ?, the current literature explains the asymptotic tail behaviour of \(M(\mathcal {T})\) under some regularity conditions including that 1 ? σ(t) has a polynomial decrease to 0 as t → t ₀. In this contribution we consider more general case that 1 ? σ(t) is regularly varying at t ₀. We extend our analysis to Gaussian random fields defined on some compact set \(\mathcal {T}\subset \mathbb {R}^{2}\), deriving the exact tail asymptotics of \(M(\mathcal {T})\) for the class of Gaussian random fields with variance and correlation functions being regularly varying at t ₀. A crucial novel element is the analysis of families of Gaussian random fields that do not possess locally additive dependence structures, which leads to qualitatively new types of asymptotics.     

Asymptotic Representation of L<Subscript>p</Subscript>-Minimal Polynomials, 1 < p < ∞

While the theory of asymptotics for L₂-minimal polynomials is highly developed, so far not much is known about L_p-minimal polynomials, Indeed, Bernstein gave asymptotics for the minimum deviation, Fekete and Walsh gave nth root asymptotics and, recently, Lubinsky and Saff came up with asymptotics outside the support [-1,1]. But the main point of interest, the asymptotic representation on the support, still remains open. Here we settle it for weight functions of the form where w is positive and on [-1,1] with and 相似文献   

Various types of stochastic integrals with respect to fractional Brownian sheet and their applications

In this paper, we develop a stochastic calculus related to a fractional Brownian sheet as in the case of the standard Brownian sheet. Let be a fractional Brownian sheet with Hurst parameters H=(H₁,H₂), and (²[0,1],B(²[0,1]),μ) a measure space. By using the techniques of stochastic calculus of variations, we introduce stochastic line integrals along all sufficiently smooth curves γ in ²[0,1], and four types of stochastic surface integrals: , i=1,2, , , , . As an application of these stochastic integrals, we prove an Itô formula for fractional Brownian sheet with Hurst parameters H₁,H₂∈(1/4,1). Our proof is based on the repeated applications of Itô formula for one-parameter Gaussian process.     

A weighted occupation time for a class of measured-valued branching processes

Summary A weighted occupation time is defined for measure-valued processes and a representation for it is obtained for a class of measure-valued branching random motions on R ^d. Considered as a process in its own right, the first and second order asymptotics are found as time t. Specifically the finiteness of the total weighted occupation time is determined as a function of the dimension d, and when infinite, a central limit type renormalization is considered, yielding Gaussian or asymmetric stable generalized random fields in the limit. In one Gaussian case the results are contrasted in high versus low dimensions.Research supported in part by Natural Sciences and Engineering Research Council of Canada     

On Riesz transforms and maximal functions in the context of Gaussian Harmonic Analysis

The purpose of this paper is twofold. We introduce a general maximal function on the Gaussian setting which dominates the Ornstein-Uhlenbeck maximal operator and prove its weak type by using a covering lemma which is halfway between Besicovitch and Wiener. On the other hand, by taking as a starting point the generalized Cauchy-Riemann equations, we introduce a new class of Gaussian Riesz Transforms. We prove, using the maximal function defined in the first part of the paper, that unlike the ones already studied, these new Riesz Transforms are weak type independently of their orders.

     

Strassen theorems for a class of iterated processes

A general direct Strassen theorem is proved for a class of stochastic processes and applied for iterated processes such as , where is a standard Wiener process and is a local time of a Lévy process independent from .

     

An exact asymptotics for small deviations of a nonstationary Ornstein-Uhlenbeck process in the L p -norm, p ≥ 2

A result concerning an exact asymptotics for the probability $P\{ \int\limits_0^1 {|\zeta _\gamma (t)|^p dt \leqslant \varepsilon ^p } \} ,\varepsilon \to 0$ , where p ≥ 2, is proved for a nonstationary Gaussian Markov process ζ _γ (t) of Ornstein-Uhlenbeck with zero mean and the covariance function $E\zeta _\gamma (t)\zeta _\gamma (s) = \tfrac{1}{{2\gamma }}[e^{ - \gamma |t - s|} - e^{ - \gamma |t + s|} ]$ , s, t ≥ 0, where γ > 0 is a parameter. Investigation techniques are the Laplace method for the sojourn time of continuous-time Markov processes and reduction to the case of Wiener processes.     

Local asymptotic mixed normality of transformed Gaussian models for random fields

Local asymptotic mixed normality (LAMN) of a class of transformed Gaussian models for discretely observed random fields is proved. The original Gaussian random field is assumed to be the product of a deterministic process and a process with independent increments. The transformed process is observed only on discrete lattice points in the unit cube and fixed domain asymptotics is investigated. This model is useful for modeling random fields with non-Gaussian marginal distributions.